Let $\omega = e^{i \pi / 3}$,and $a, b, c, x, y, z$ be non-zero complex numbers such that $a+b+c = x$,$a+b \omega + c \omega^2 = y$,and $a+b \omega^2 + c \omega = z$. Then the value of $\frac{|x|^2+|y|^2+|z|^2}{|a|^2+|b|^2+|c|^2}$ is

Let $a, b$ be two real numbers such that $ab < 0$. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$,then a possible value of $\frac{1+[a]}{4b}$,where $[t]$ is the greatest integer function,is:

Consider the regions for complex number $z$ defined by $A: \frac{1}{\log_2 |z|} - \frac{1}{\log_2 |z| - 1} - 1 < 0$ and $B: \operatorname{Im}(z) = 0$. The range of values of $\operatorname{Re}(z)$ lying in the region $A \cap B$ is

If $(1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni) = a + ib$,then $2 \times 5 \times 10 \times \dots \times (1 + n^2)$ is equal to

For a non-zero complex number $z$,let $\arg(z)$ denote the principal argument with $-\pi < \arg(z) \leq \pi$. Then,which of the following statement$(s)$ is (are) $FALSE$?
$(A)$ $\arg(-1-i) = \frac{\pi}{4}$,where $i = \sqrt{-1}$
$(B)$ The function $f: \mathbb{R} \rightarrow (-\pi, \pi]$,defined by $f(t) = \arg(-1+it)$ for all $t \in \mathbb{R}$,is continuous at all points of $\mathbb{R}$,where $i = \sqrt{-1}$
$(C)$ For any two non-zero complex numbers $z_1$ and $z_2$,$\arg\left(\frac{z_1}{z_2}\right) - \arg(z_1) + \arg(z_2)$ is an integer multiple of $2\pi$.
$(D)$ For any three given distinct complex numbers $z_1, z_2$ and $z_3$,the locus of the point $z$ satisfying the condition $\arg\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right) = \pi$ lies on a straight line.

For a complex number $z$, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all complex numbers $z$ satisfying $z^4 - |z|^4 = 4iz^2$, where $i = \sqrt{-1}$. Then the minimum possible value of $|z_1 - z_2|^2$, where $z_1, z_2 \in S$ with $\operatorname{Re}(z_1) > 0$ and $\operatorname{Re}(z_2) < 0$, is: